InfoPath Dev

I have a form that is a general form that needs to go into multiple SharePoint 2010 libraries on multiple subsites. I have multiple sites for various customers that all use the same form but all of these sites are within the same site collection. How can I go about uploading this form so that multiple form libraries can use it while also, when they submit it, it submits a copy in their form library. I basically want to publish the same form to multiple libraries to use as their default template and have the ability to easily update one file in one location and have all the forms update. If I create a template and then just set it as the default template in certain libraries will those libraries be smart enough to save a copy inside of themselves? Hopefully my question makes sense and someone can help out. Thanks in advance for any help.

To clarify a little bit, I am able to publish any form as a site content type and then enable it for a Form Library. What I want to have happen is, when I set it as the default content type, I want whatever Form Library it is in, to retain a copy of the submitted form. Basically to submit to the library in which it resides. But I don't want to have to set this manually for each library that I set the content type for.